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Chapter 11: Problem 52

Describe the level curves of the function. Sketch the level curves for the given \(c\) -values. $$ f(x, y)=x^{2}+2 y^{2}, \quad c=0,2,4,6,8 $$

Short Answer

Expert verified
The level curves of the function \(f(x, y)=x^{2}+2 y^{2}\) for values \(c = 0, 2, 4, 6, 8\) correspond to a point at the origin and a series of increasingly larger ellipses with the same center, growing as the value of \(c\) grows. Therefore, the level curves represent circular paths around the origin.

Step by step solution

01

Formulate the General Equation for the Level Curves

Retain the general structure of the function \(f(x, y)=x^{2}+2 y^{2}\), setting it equal to the constant \(c\), which represents the 'height' of our three-dimensional surface. So, we have \(x^{2}+2 y^{2} = c\). This is our general equation, which we can solve for different values of \(c\).
02

Construct Level Curves for Different Values of \(c\)

Substitute each of the given \(c\)-values, \(0,2,4,6,8\), into the level curve equation from the first step. For \(c = 0\), we get the equation \(x^2+2y^2 = 0\). For \(c = 2\), our equation is \(x^2+2y^2 = 2\), and so on.
03

Transform Each Equation into a Recognizable Form

Rework each equation into a form that can be easily recognized and graphed. In general, the equation \(x^2 + ay^2 = c\) describes an ellipse for \(a > 0\) and \(c > 0\). For \(c = 0\), the equation describes a single point at the origin. Our equations thus represent ellipses with the same center but different radii. The equation for \(c = 2\) for example can be simplified to \(x^2 + y^2 = 1\) and \(2y^2 = 1\), or \(x^2 + (\sqrt{2}y)^2 = 1\), an ellipse with semi-major axis \(1\) and semi-minor axis \(\frac{1}{\sqrt{2}}\), and so on for other \(c\)-values.
04

Sketch the Level Curves

Using the results from Step 3, sketch each level curve on a separate graph. Notice that as the value of \(c\) increases, the corresponding level curves become larger. With \(c = 0\), the curve is a single point at the origin. With \(c = 2\), the curve is an ellipse with semi-major axis \(1\) and semi-minor axis \(\frac{1}{\sqrt{2}}\). The size of the ellipse grows for following values of \(c\) until the largest ellipse corresponding to the value \(c = 8\).

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